Sketch the graphs of the given functions on the same axes. y=((1)/(2))^x, y=((1)/(3))^x, and y

Sketch the graphs of the given functions on the same axes....

Key Concepts

Exponential Functions

Exponential functions are functions of the form y = a^x, where the constant a is the base. In this context, all functions have bases between 0 and 1, which characterizes them as exponential decay functions. Exponential functions are important in modeling processes that decrease or grow rapidly and are recognized by their constant growth rate on a multiplicative scale.

Rate of Decay

The rate of decay in an exponential function determines how quickly the function values approach zero as x increases. A smaller base (closer to zero) results in a faster decay, whereas a base closer to one decays more slowly. This concept is important for comparing different exponential functions, as even slight differences in the base can significantly change how steeply the graph decreases.

Horizontal Asymptote

All exponential decay functions with positive bases have a horizontal asymptote at y = 0. This means that as x tends to infinity, the function values approach zero but never actually reach it. Recognizing the horizontal asymptote helps in graphing by providing a reference line that the function approaches, thereby indicating the long-term behavior of the functions.

Transcript

00:01 In this question three functions are given that is y equals to 1 by 2 raise to the power x x, y equals to 1 by 3 raised to the power x and y equals to 1 by 4 raised to the power x.

00:12 So we have to sketch the graph of the given function on same axis.

00:15 So no problem.

00:16 First of all we will take some properties of these graphs.

00:20 So first of all all all three functions are defined for real number, no problem, okay.

00:25 And they all have y intercept at x equals to 0.

00:29 When x equals to 0 y will be 1 here y will be 1 and y will be 1 okay so i'm writing down here for x equals to 0 y will be 1 and no x intercept okay and we can see the base is here in all three functions less than 1 so it is less than 1 so we can say these graphs will be decreasing okay and now we will take some points okay to take some values for the function which will help us to sketch the graph so we can take here x here 1 by 2 raise to the power x 1 by 3 raised to the power x and 1 by 4 raise to the power x okay so when we take we will take some points x equals to minus 3 minus 2 minus 1 0 1 2 and 3 okay so 1 by 2 raise to the power x and x is negative 3 that will be 4 okay 1 by 2 raise to the power negative 2 that is 4 1 by 2 raise to the power minus 1 that is 2 okay and same it will be x equals to 0 it will be 1 and same it will be two sorry it will be one by two one by four and one by eight okay and now we will do same from here it will be 27 and nine and three okay and it will be one now one divided by all threes so it will be one by three one by nine and one by 27 and same for one by four raised to the power x it will be 64 okay and 16 and four it will be one and 1 by 4 1 by 16 and 1 by 64 so we can see as the value of x increasing the functions are decreasing okay you can see because the base is here less than 1 so now we will sketch the graph so i have made this graph for you okay that is i will show you here okay that is like this so this is the graph, okay, you can see in the green color that is 1 by 4 raised to the power x, okay? 1 by 4 raised to the power x...

Please give Ace some feedback